A random vector is a finite-dimensional formal vector of random variables. The random vector can be written either as a column or row of random variables, depending on its context and use. So if are random variables, then
is a random (column) vector. Similarly, one defines a random matrix to be a formal matrix whose entries are all random variables. The size of a random vector and the size of a random matrix are assumed to be finite fixed constants.
Similarly, the distribution of a random matrix is the joint distribution of its matrix components.
Let be a random vector. If exists () for each , then the expectation of X, called the mean vector and denoted by , is defined to be:
Clearly . The expectation of a random matrix is similarly defined. Note that the definitions of expectations can also be defined via measure theory. Then, using Fubini’s Theorem, one can show that the two sets of definitions coincide.
Again, let be a random vector. If = is defined and are defined for all , then the variance of X, denoted by , is defined to be:
If X is an -dimensional random vector with A a constant matrix and an -dimensional constant vector, then
Same set up as above. Then
If the ’s are iid (independent identically distributed), with variance , then
Let be an -dimensional random vector with , . is an constant matrix. Then
|Date of creation||2013-03-22 14:27:20|
|Last modified on||2013-03-22 14:27:20|
|Last modified by||CWoo (3771)|
|Defines||distribution of a random vector|
|Defines||distribution of a random matrix|